Can I use something similar to "Slide Vertices" to create new vertices and edges?
Here is a related question: Duplicate and slide edge (rip from boundary)
I don't think there is a way to do exactly what you want in vanilla blender. But there are a few things you can try depending on what you want to achieve exactly, and what kind of mesh you're working on.

Just for a quick reference, instead of sliding verts with Shift-V, you can do the same more quickly by just pressing GG while your vertices or edges are selected.

The first thing that comes to mind is the Offset Edge Slide tool. Not to be confused with normal Edge Slide. All you have to do is select the vertices you want, and then press CTRL + SHIFT + R. This will slide your new vertices where you want, but as a downside it will do so symmetrically. So you will have to clean up the side that you don't want by just dissolving.


Here's an example video:



Another option that could work how you describe is just a simple extrude. The key is to change the snapping options to edge, and when you are extruding hold CTRL and hover over the desired edge. Once you've completed the extrusion, you will need to merge duplicate vertices by pressing M and selecting &quot;By Distance&quot; since this method will create duplicate geometry.

Example video:


Hope this helps.